Calculate the value of A, if :
`(tanA-1)(cosec3A-1)=0`

Answer

To solve the equation \((\tan A - 1)(\csc 3A - 1) = 0\), we can break it down into two separate equations:

1. \(\tan A - 1 = 0\)
2. \(\csc 3A - 1 = 0\)

### Step 1: Solve \(\tan A - 1 = 0\)

From the first equation, we can set:

\[ \tan A = 1 \]

The tangent function equals 1 at specific angles. In degrees, we know:

\[ \tan 45^\circ = 1 \]

Thus, we have:

\[ A = 45^\circ \]

### Step 2: Solve \(\csc 3A - 1 = 0\)

From the second equation, we can set:

\[ \csc 3A = 1 \]

The cosecant function equals 1 when its reciprocal, the sine function, equals 1. Therefore:

\[ \sin 3A = 1 \]

The sine function equals 1 at:

\[ 3A = 90^\circ + n \cdot 360^\circ \quad (n \in \mathbb{Z}) \]

Dividing by 3 gives:

\[ A = 30^\circ + n \cdot 120^\circ \]

For \(n = 0\):

\[ A = 30^\circ \]

For \(n = 1\):

\[ A = 30^\circ + 120^\circ = 150^\circ \]

For \(n = 2\):

\[ A = 30^\circ + 240^\circ = 270^\circ \]

### Summary of Solutions

The possible values of \(A\) are:

1. \(A = 45^\circ\)
2. \(A = 30^\circ\)
3. \(A = 150^\circ\)
4. \(A = 270^\circ\)

### Final Answer

The values of \(A\) are \(45^\circ\) and \(30^\circ\).

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